Weierstrass's elliptic functions

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In mathematics, Weierstrass introduced some particular elliptic functions that have become the basis for the most standard notations used.

Contents

Definitions

Consider two complex numbers <math>\omega_1<math> and <math>\omega_2<math> defining a lattice. There are some significant choices of convention, and the literature is not consistent in its usage. It is common to name these constants so that <math>\omega_2/\omega_1<math> has a positive imaginary part. As defined below, the two numbers serve as half-periods. Compare the trigonometric usage of 2π.

Then Weierstrass's elliptic function is an elliptic function with periods <math>2\omega_1<math> and <math>2\omega_2<math> is defined as

<math>

\wp(z;\omega_1,\omega_2)=\frac{1}{z^2}+ \sum{}' \left\{ \frac{1}{(z-2m\omega_1-2n\omega_2)^2}- \frac{1}{\left(2m\omega_1+2n\omega_2\right)^2} \right\} <math>

where <math>\sum{}'<math> represents the sum over all pairs of integers <math>m<math> and <math>n<math> except <math>m=n=0<math>. It is usual to write <math>\Omega_{m,n}=2m\omega_1+2n\omega_1<math>, the points of the period lattice, so that

<math>\wp(z;\omega_1,\omega_2)=

z^{-2}+\sum{}'\left\{(z-\Omega_{m,n})^{-2}-\Omega_{m,n}^{-2} \right\}<math>. There is thus a second order pole at each point of the period lattice (including the origin). With these definitions, <math>\wp(z)<math> is an even function and its derivative <math>\wp'<math> an odd function.

Further development of the theory of elliptic functions shows that the condition on Weierstrass's function (correctly called pe) is determined up to addition of a constant and multiplication by a non-zero constant by the condition on the poles alone, amongst all meromorphic functions with the given period lattice.

It can be shown that

<math>

\wp(z;\omega_1,\omega_2)= \left(\frac{\pi}{2\omega_1}\right)^2\left[ -\frac{1}{3}+\sum_{n=-\infty}^{n=+\infty}{\rm cosec}^2\left(\frac{z-2n\omega_2}{2\omega_1}\pi\right)- \sum_{n=-\infty}^{n=+\infty}{}'{\rm cosec}^2\frac{n\omega_2}{\omega_1}\pi\right],<math> which converges faster than the other formula given above.

Invariants

Missing image
Gee_three_real.jpeg
The real part of the invariant g_3 as a function of the nome q on the unit disk.
Missing image
Gee_three_imag.jpeg
The imaginary part of the invariant g_3 as a function of the nome q on the unit disk.

If points close to the origin are considered the appropriate Laurent series is

<math>

\wp(z;\omega_1,\omega_2)=z^{-2}+\frac{1}{20}g_2z^2+\frac{1}{28}g_3z^4+O(z^6) <math> where

<math>g_2= 60\sum{}' \Omega_{m,n}^{-4},\qquad
      g_3=140\sum{}' \Omega_{m,n}^{-6}.<math>

The numbers <math>g_2<math> and <math>g_3<math> are known as the invariants — they are two terms out of the Eisenstein series. (Abramowitz and Stegun restrict themselves to the case of real <math>g_2<math> and <math>g_3<math>, stating that this case "seems to cover most applications"; this may be true from the point of view of applied mathematics. If <math>\omega_1<math> is real and <math>\omega_2<math> pure imaginary, or if <math>\omega_1=\overline{\omega_2}<math>, the invariants are real).

Note that <math>g_2<math> and <math>g_3<math> are homogeneous functions of degree -4 and -6; that is,

<math>g_2(\lambda \omega_1, \lambda \omega_2) = \lambda^{-4} g_2(\omega_1, \omega_2)<math>

and

<math>g_3(\lambda \omega_1, \lambda \omega_2) = \lambda^{-6} g_3(\omega_1, \omega_2)<math>.

Thus, by convention, one frequently writes <math>g_2<math> and <math>g_3<math> in terms of the half-period ratio <math>\tau=\omega_2/\omega_1<math> and take <math>\tau<math> to lie in the upper half plane. Thus, <math>g_2(\tau)=g_2(1, \omega_2/\omega_1)<math> and <math>g_3(\tau)=g_3(1, \omega_2/\omega_1)<math>.

The Fourier series for <math>g_2<math> and <math>g_3<math> can be written in terms of the square of the nome <math>q=\exp(i\pi\tau)<math> as

<math>g_2(\tau)=\frac{4\pi^4}{3} \left[ 1+ 240\sum_{k=1}^\infty \sigma_3(k) q^{2k} \right] <math>

and

<math>g_3(\tau)=\frac{8\pi^6}{27} \left[ 1- 504\sum_{k=1}^\infty \sigma_5(k) q^{2k} \right] <math>

where <math>\sigma_a(k)<math> is the divisor function. This formula may be re-written in terms of Lambert series.

The invariants may be expressed in terms of Jacobi's theta functions. This method is very convenient for numerical calculation: the theta functions converge very quickly. In the notation of Abramowitz and Stegun, but denoting the primitive half-periods by <math>\omega_1,\omega_2<math>, the invariants satisfy

<math>

g_2(\omega_1,\omega_2)= \frac{\pi^4}{12\omega_1^4} \left(

   \theta_2(0,q)^8-\theta_3(0,q)^4\theta_2(0,q)^4+\theta_3(0,q)^8

\right) <math> and

<math>

g_3(\omega_1,\omega_2)= \frac{\pi^6}{(2\omega_1)^6} \left[

  \frac{8}{27}\left(\theta_2(0,q)^{12}+\theta_3(0,q)^{12}\right)\right.

<math>

<math>\left. -
  \frac{4}{9}\left(\theta_2(0,q)^4+\theta_3(0,q)^4\right)\cdot
             \theta_2(0,q)^4\theta_3(0,q)^4

\right] <math> where <math>\tau=\omega_2/\omega_1<math> is the half-period ratio and <math>q=e^{\pi i\tau}<math> is the nome.

Special cases

If the invariants are <math>g_2=0<math>, <math>g_3=1<math>, then this is known as the Equianharmonic case; <math>g_2=1<math>, <math>g_3=0<math> is the Lemniscatic case.

Differential equation

With this notation, the <math>\wp<math> function satisfies the following differential equation:

<math>

[\wp'(z)]^2=4[\wp(z)]^3-g_2\wp(z)-g_3,<math> where dependence on <math>\omega_1<math> and <math>\omega_2<math> is suppressed.

Integral equation

The Weierstrass elliptic function can be given as the inverse of an elliptic integral. Let

<math>u = \int_y^\infty \frac {ds} {\sqrt{4s^3 - g_2s -g_3}}<math>.

Here, g2 and g3 are taken as constants. Then one has

<math>y=\wp(u)<math>.

The above follows directly by integrating the differential equation.

Modular discriminant

Missing image
Discriminant_real_part.jpeg
The real part of the discriminant as a function of the nome q on the unit disk.

The modular discriminant <math>\Delta<math> is defined as

<math>

\Delta=g_2^3-27g_3^2.<math>

This is studied in its own right, as a cusp form, in modular form theory (that is, as a function of the period lattice).

Note that <math>\Delta=(2\pi)^{12}\eta^{24}<math> where <math>\eta<math> is the Dedekind eta function.

The discriminant is a modular form of weight 12. That is, under the action of the modular group, it transforms as

<math>\Delta \left( \frac {a\tau+b} {c\tau+d}\right) =

\left(c\tau+d\right)^{12} \Delta(\tau)<math> with τ being the half-period ratio, and a,b,c and d being integers, with ad-bc=1.

The constants e1, e2 and e3

Consider the algebraic equation <math>4t^3-g_2t-g_3=0<math>, and name its roots <math>e_1<math>, <math>e_2<math>, and <math>e_3<math>. It can be shown from the non-vanishing of the discriminant that no two of these three are equal.

Algebraic considerations show that <math>e_1+e_2+e_3=0<math>.

In the case of real invariants, the sign of <math>\Delta<math> determines the nature of the roots. If <math>\Delta>0<math>, all three are real and it is conventional to name them so that <math>e_1>e_2>e_3<math>. If <math>\Delta<0<math>, it is conventional to write <math>e_1=-\alpha+\beta i<math> (where <math>\alpha\geq 0<math>, <math>\beta>0<math>), whence <math>e_3=\overline{e_1}<math> and <math>e_2<math> is real and non-negative. We also have

<math>

\wp(\omega_1)=e_1\qquad \wp(\omega_2)=e_2\qquad \wp(\omega_3)=e_3 <math> where <math>\omega_3=-\omega_1-\omega_2<math>. Also, <math>\wp'(\omega_i)=0<math> for <math>i=1,2,3<math>.

If <math>g_2<math> and <math>g_3<math> are real and <math>\Delta>0<math>, the <math>e_i<math> are all real, and <math>\wp()<math> is real on the perimeter of the rectangle with corners <math>0<math>, <math>\omega_3<math>, <math> \omega_1+\omega_3<math>, and <math>\omega_1<math>.

Addition theorems

The Weierstrass elliptic functions have several properties that may be proved:

<math>

\det\begin{bmatrix} \wp(z) & \wp'(z) & 1\\ \wp(y) & \wp'(y) & 1\\ \wp(z+y) & -\wp'(z+y) & 1 \end{bmatrix}=0<math>

(a symmetrical version would be

<math>

\det\begin{bmatrix} \wp(u) & \wp'(u) & 1\\ \wp(v) & \wp'(v) & 1\\ \wp(w) & -\wp'(w) & 1 \end{bmatrix}=0<math> where <math>u+v+w=0<math>).

Also

<math>

\wp(z+y)=\frac{1}{4} \left\{ \frac{\wp'(z)-\wp'(y)}{\wp(z)-\wp(y)} \right\}^2 -\wp(z)-\wp(y).<math>

and the duplication formula

<math>

\wp(2z)= \frac{1}{4}\left\{ \frac{\wp''(z)}{\wp'(z)}\right\}^2-2\wp(z),<math> unless <math>2z<math> is a period.

The case with 1 a basic half-period

If <math>\omega_1=1<math>, much of the above theory becomes simpler; it is then conventional to write <math>\tau<math> for <math>\omega_2<math>. For a fixed τ in the upper half plane, so that the imaginary part of τ is positive, we define the Weierstrass <math>\wp<math> function by:

<math>\wp(z;\tau) =\frac{1}{z^2} + \sum_{n^2+m^2 \ne 0}{1 \over (z-n-m\tau)^2} - {1 \over (n+m\tau)^2}<math>

The sum extends over the lattice {n+mτ : n and m in Z} with the origin omitted. Here we regard τ as fixed and <math>\wp<math> as a function of <math>z<math>; fixing <math>z<math> and letting τ vary leads into the area of elliptic modular functions.

General theory

<math>\wp<math> is a meromorphic function in the complex plane with poles at the lattice points. It is doubly periodic with periods 1 and τ; this means that <math>\wp<math> satisfies

<math>\wp(z+1) = \wp(z+\tau) = \wp(z)<math>

The above sum is homogeneous of degree minus two, and if <math>c<math> is any non-zero complex number,

<math>\wp(cz;c\tau) = \wp(z;\tau)/c^2<math>

from which we may define the Weierstrass <math>\wp<math> function for any pair of periods. We also may take the derivative (of course, with respect to z) and obtain a function algebraically related to <math>\wp<math> by

<math>\wp'^2 = \wp^3 - g_2 \wp - g_3<math>

where <math>g_2<math> and <math>g_3<math> depend only on τ, being modular forms. The equation

<math>Y^2 = X^3 - g_2 X - g_3<math>

defines an elliptic curve, and we see that (<math>\wp<math>,<math>\wp'<math>) is a parametrization of that curve.

The totality of meromorphic doubly periodic functions with given periods defines an algebraic function field, associated to that curve. It can be shown that this field is

<math>\Bbb{C}(\wp, \wp')<math>,

so that all such functions are rational functions in the Weierstrass function and its derivative.

We can also wrap a single period parallelogram into a torus, or donut-shaped Riemann surface, and regard the elliptic functions associated to a given pair of periods to be functions defined on that Riemann surface.

The roots <math>e_1<math>, <math>e_2<math>, and <math>e_3<math> of the equation <math>X^3 - g_2 X - g_3<math> depend on τ and can be expressed in terms of theta functions; we have

<math>e_1(\tau) = {\pi^2 \over {3}}(\vartheta^4(0;\tau) + \vartheta_{01}^4(0;\tau))<math>
<math>e_2(\tau) = -{\pi^2 \over {3}}(\vartheta^4(0;\tau) + \vartheta_{10}^4(0;\tau))<math>
<math>e_3(\tau) = {\pi^2 \over {3}}(\vartheta_{10}^4(0;\tau) - \vartheta_{01}^4(0;\tau))<math>

Since <math>g_2 = -4(e_1e_2+e_2e_3+e_3e_1)<math> and <math>g_3 = 4e_1e_2e_3<math> we have these in terms of theta functions also.

We may also express <math>\wp<math> in terms of theta functions; because these converge very rapidly, this is a more expeditious way of computing <math>\wp<math> than the series we used to define it.

<math>\wp(z; \tau) = \pi^2 \vartheta^2(0;\tau) \vartheta_{10}^2(0;\tau){\vartheta_{01}^2(z;\tau) \over \vartheta_{11}^2(z;\tau)} + e_2(\tau)<math>

The function <math>\wp<math> has two zeroes (modulo periods) and the function <math>\wp'<math> has three. The zeroes of <math>\wp'<math> are easy to find: since <math>\wp'<math> is an odd function they must be at the half-period points. On the other hand it is very difficult to express the zeroes of <math>\wp<math> by closed formula, except for special values of the modulus (e.g. when the period lattice is the Gaussian integers). An expression was found, by Zagier and Eichler.

The Weierstrass theory also includes the Weierstrass zeta function, which is an indefinite integral of <math>\wp<math> and not doubly-periodic, and a theta function called the Weierstrass sigma function, of which his zeta-function is the log-derivative. The sigma-function has zeroes at all the period points (only), and can be expressed in terms of Jacobi's functions. This gives one way to convert between Weierstrass and Jacobi notations.

The Weierstrass sigma-function is an entire function; it played the role of 'typical' function in a theory of random entire functions of J. E. Littlewood.

References

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